Summary: Extensional Equality in Intensional Type Theory
Thorsten Altenkirch
Department of Informatics
University of Munich
Oettingenstr. 67, 80538 M¨unchen, Germany,
alti@informatik.uni-muenchen.de
Abstract
We present a new approach to introducing an extensional
propositional equality in Intensional Type Theory. Our con-
struction is based on the observation that there is a sound,
intensional setoid model in Intensional Type theory with a
proof-irrelevant universe of propositions and
-rules for ¡ -
and ¢ -types. The Type Theory corresponding to this model
is decidable, has no irreducible constants and permits large
eliminations, which are essential for universes.
Keywords. Type Theory, categorical models.
1. Introduction and Summary
In Intensional Type Theory (see e.g. [11]) we differen-
tiate between a decidable definitional equality (which we